6 Conclusions
Causal inference in time series settings is a challenging task, partly because a simple associ-ation may be easily mistaken for a causal nexus and partly due to serial dependence, which brings additional difficulties during the estimation process. Moreover, in a panel setting where a treatment assigned to some units affects the others (a situation known as “interference”), inferring a causal effect is particularly demanding.
Having its roots in the potential outcomes framework, we believe that the RCM can aid the estimation of causal effects in such settings. Indeed, the RCM allows the construction of “what if” scenarios and sets the theoretical foundations underneath the attribution of the uncovered effect to a specific intervention. In particular, in this research we analyzed three situations that researchers and practitioners commonly encounter in a time series analysis: i) a single intervention occurring simultaneously on multiple non-interfering series; ii) multiple time series subject to a simultaneous treatment that, due to cross-unit interactions, may affect other series that were not intervened on; iii) multiple interventions on a single time series.
We first introduced a common causal framework building the theoretical foundations for a causal analysis under the RCM; we then defined new classes of causal estimands in each of the three settings and we proposed to estimate them using two novel methodologies: C-ARIMA and CausalMBSTS. The C-ARIMA approach can successfully estimate the effect of an intervention on a single time series as well as on multiple non-interfering series. Indeed, with a simulation study we showed that it performs well compared to a standard intervention analysis method in a situation where the effect takes the form of a fixed change in the level of the outcome; it also outperforms the latter when the effects are irregular and time-varying. Instead, CausalMBSTS can estimate the heterogeneous causal effect of an intervention in a panel setting where the time series interact with one another. Based on multivariate Bayesian models, it is a flexible methodology that allows to model the dependence structure between the time series in a very natural way, whilst enabling variable selection (via the addition of a spike-and-slab prior) and validation (by posterior predictive checks). Finally, we showed how to extend the C-ARIMA approach for the estimation of the heterogeneous causal effects of multiple interventions occurring on a single time series. We also applied the proposed C-ARIMA and CausalMBSTS approaches to evaluate the effect of a permanent price reduction introduced by a supermarket chain in Italy on a selected subset of store brands. Then, we used C-ARIMA in a multi-intervention setting to investigate the impact of the first two regulated Bitcoin futures on Bitcoin volatility and daily transaction volumes.
This research brings both methodological and empirical contributions, introducing two novel approaches to infer causal effects in complex time series settings and showing that the proposed methodologies can be employed in several fields of research, including marketing and finance.
We also believe that this research provides several advances to the existing inferential methods
under the RCM. Indeed, recent extensions of the RCM to observational panel studies (i.e., synthetic control methods) mostly focus on a situation where the time series do not interact with one another and they also involve inferential tools that are not usually employed in stan-dard time series analysis. Conversely, the C-ARIMA approach allows the estimation of causal effects under the RCM with standard tools that are extensively used by econometricians and practitioners in many fields. Furthermore, with CausalMBSTS we extended synthetic control methods to a setting with interference and, to foster the adoption of this method by a broad range of researchers, we also released an R package that handles both the definition and the estimation of the multivariate model. By making causal inference tools accessible to a vast audience, we hope we can facilitate the understanding of causal relationships and, in turn, decision making based on solid foundations.
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References
Abadie, A. (2005). Semiparametric difference-in-differences estimators. The Review of Eco-nomic Studies, 72(1):1–19.
Abadie, A., Diamond, A., and Hainmueller, J. (2010). Synthetic control methods for compar-ative case studies: Estimating the effect of california’s tobacco control program. Journal of the American Statistical Association, 105(490):493–505.
Abadie, A., Diamond, A., and Hainmueller, J. (2015). Comparative politics and the synthetic control method. American Journal of Political Science, 59(2):495–510.
Abadie, A. and Gardeazabal, J. (2003). The economic costs of conflict: A case study of the basque country. American economic review, 93(1):113–132.
Anger, S., Kvasnicka, M., and Siedler, T. (2011). One last puff? public smoking bans and smoking behavior. Journal of health economics, 30(3):591–601.
Angrist, J. D. and Pischke, J.-S. (2008). Mostly harmless econometrics: An empiricist’s com-panion. Princeton university press.
Antoniou, A. and Holmes, P. (1995). Futures trading, information and spot price volatility:
evidence for the FTSE-100 stock index futures contract using GARCH. Journal of Banking
& Finance, 19(1):117–129.
Antoniou, A., Holmes, P., and Priestley, R. (1998). The effects of stock index futures trading on stock index volatility: An analysis of the asymmetric response of volatility to news. Journal of Futures Markets: Futures, Options, and Other Derivative Products, 18(2):151–166.
Athey, S. and Imbens, G. W. (2018). Design-based analysis in difference-in-differences settings with staggered adoption. Technical report, National Bureau of Economic Research.
Baklaci, H. and Tutek, H. (2006). The impact of the futures market on spot volatility: An analysis in Turkish derivatives markets. WIT Transactions on Modelling and Simulation, 43:237–246.
Balke, N. S. and Fomby, T. B. (1994). Large shocks, small shocks, and economic fluctuations:
outliers in macroeconomic time series. Journal of Applied Econometrics, 9(2):181–200.
Basse, G. W., Feller, A., and Toulis, P. (2019). Randomization tests of causal effects under interference. Biometrika, 106(2):487–494.
Baydakova, A. (2019). No change to bitcoin futures plans, CME says, as CBOE pulls back.
Coindesk, available at https://www.coindesk.com/cme-cboe-bitcoin-futures.
Ben-Michael, E., Feller, A., and Rothstein, J. (2018). The augmented synthetic control method.
Preprint. Available atarXiv:1811.04170.
Ben-Michael, E., Feller, A., and Rothstein, J. (2019). Synthetic controls and weighted event studies with staggered adoption. Preprint. Available atarXiv: 1912. 03290.
Bessembinder, H. and Seguin, P. J. (1992). Futures-trading activity and stock price volatility.
the Journal of Finance, 47(5):2015–2034.
Bhattacharyya, M. and Layton, A. P. (1979). Effectiveness of seat belt legislation on the queensland road toll—an Australian case study in intervention analysis. Journal of the American Statistical Association, 74(367):596–603.
Billmeier, A. and Nannicini, T. (2013). Assessing economic liberalization episodes: A synthetic control approach. Review of Economics and Statistics, 95(3):983–1001.
Blattberg, R. C., Briesch, R., and Fox, E. J. (1995). How promotions work. Marketing science, 14:G122–G132.
Bohl, M. T., Diesteldorf, J., and Siklos, P. L. (2015). The effect of index futures trading on volatility: Three markets for chinese stocks. China Economic Review, 34:207–224.
Bojinov, I., Rambachan, A., and Shephard, N. (2020). Panel experiments and dynamic causal effects: A finite population perspective. Preprint. Available at arXiv:2003.09915.
Bojinov, I. and Shephard, N. (2019). Time series experiments and causal estimands: exact ran-domization tests and trading. Journal of the American Statistical Association, 114(528):1665–
1682.
Borenstein, M., Hedges, L. V., Higgins, J. P., and Rothstein, H. R. (2011). Introduction to meta-analysis. John Wiley & Sons.
Bouoiyour, J. and Selmi, R. (2015). What does Bitcoin look like? Annals of Economics &
Finance, 16(2):449–492.
Box, G. E. and Tiao, G. C. (1975). Intervention analysis with applications to economic and environmental problems. Journal of the American Statistical association, 70(349):70–79.
Box, G. E. and Tiao, G. C. (1976). Comparison of forecast and actuality. Journal of the Royal Statistical Society: Series C (Applied Statistics), 25(3):195–200.
Brodersen, K. H., Gallusser, F., Koehler, J., Remy, N., and Scott, S. L. (2015). Inferring causal impact using bayesian structural time-series models. The Annals of Applied Statistics, 9(1):247–274.
82
Brown, P. J., Vannucci, M., and Fearn, T. (1998). Multivariate bayesian variable selection and prediction. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 60(3):627–641.
Cao, J. and Dowd, C. (2019). Estimation and inference for synthetic control methods with spillover effects. Preprint. Available at arXiv:1902.07343.
Card, D. and Krueger, A. B. (1993). Minimum wages and employment: A case study of the fast food industry in new jersey and pennsylvania. Technical report, National Bureau of Economic Research.
Cauley, J. and Im, E. I. (1988). Intervention policy analysis of skyjackings and other terrorist incidents. The American Economic Review, 78(2):27–31.
Cermak, V. (2017). Can Bitcoin become a viable alternative to fiat currencies? An empirical analysis of bitcoin’s volatility based on a GARCH model. Available athttps: // ssrn. com/
abstract= 2961405.
Chamberlain, G. (1982). The general equivalence of granger and sims causality. Econometrica:
Journal of the Econometric Society, 50(3):569–581.
Chen, C. and Liu, L.-M. (1993). Joint estimation of model parameters and outlier effects in time series. Journal of the American Statistical Association, 88.
Chen, H., Han, Q., Li, Y., and Wu, K. (2013). Does index futures trading reduce volatility in the Chinese stock market? A panel data evaluation approach. Journal of Futures Markets, 33(12):1167–1190.
Ciaian, P., Rajcaniova, M., and Kancs, A. (2016). The economics of Bitcoin price formation.
Applied Economics, 48(19):1799–1815.
Cipollini, F., Gallo, G. M., and Palandri, A. (2020). Realized variance modeling: decoupling forecasting from estimation. Journal of Financial Econometrics, 18(3):532–555.
Corbet, S., Lucey, B., Peat, M., and Vigne, S. (2018). Bitcoin futures—what use are they?
Economics Letters, 172:23–27.
Corsi, F. (2009). A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics, 7(2):174–196.
Cox, D. R. (1958). Planning of experiments. Wiley.
CryptoCompare (2020a). Exchange Benchmark Report July 2020. CryptoCompare Research, Available at: https://www.cryptocompare.com/external/research/exchange-review/.
CryptoCompare (2020b). Exchange Review 2020. CryptoCompare Research, Available at:
https://www.cryptocompare.com/external/research/exchange-review/.
Danthine, J.-P. (1978). Information, futures prices, and stabilizing speculation. Journal of Economic Theory, 17(1):79–98.
Dawid, A. P. (1981). Some matrix-variate distribution theory: notational considerations and a bayesian application. Biometrika, 68(1):265–274.
Dube, A. and Zipperer, B. (2015). Pooling multiple case studies using synthetic controls: An application to minimum wage policies. IZA Discussion Paper 8944.
Durbin, J. and Koopman, S. J. (2002). A simple and efficient simulation smoother for state space time series analysis. Biometrika, 89(3):603–616.
Edwards, F. R. (1988). Does futures trading increase stock market volatility? Financial Analysts Journal, 44(1):63–69.
Enders, W. and Sandler, T. (1993). The effectiveness of antiterrorism policies: A Vector-AutoRegression-intervention analysis. The American Political Science Review, 87(4):829–
844.
Figlewski, S. (1981). Futures trading and volatility in the GNMA market. The Journal of Finance, 36(2):445–456.
Forastiere, L., Airoldi, E. M., and Mealli, F. (2020). Identification and estimation of treat-ment and interference effects in observational studies on networks. Journal of the American Statistical Association.
Garman, M. B. and Klass, M. J. (1980). On the estimation of security price volatilities from historical data. Journal of business, 53(1):67–78.
Garvey, G. T. and Hanka, G. (1999). Capital structure and corporate control: The effect of antitakeover statutes on firm leverage. The Journal of Finance, 54(2):519–546.
Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013).
Bayesian data analysis. CRC press.
Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In Bernardo, J., Berger, J., Dawid, A., and Smith, A., editors, Bayesian Statistics 4, Oxford, UK. Clarendon Press.
Gobillon, L. and Magnac, T. (2016). Regional policy evaluation: Interactive fixed effects and synthetic controls. Review of Economics and Statistics, 98(3):535–551.
84
Granger, C. W. (1969). Investigating causal relations by econometric models and cross-spectral methods. Econometrica: journal of the Econometric Society, 37(3):424–438.
Grossi, G., Lattarulo, P., Mariani, M., Mattei, A., and ¨Oner, ¨O. (2020). Synthetic control group methods in the presence of interference: The direct and spillover effects of light rail on neighborhood retail activity. Preprint. Available at arXiv:2004.05027.
Hale, G., Krishnamurthy, A., Kudlyak, M., Shultz, P., et al. (2018). How futures trading changed Bitcoin prices. FRBSF Economic Letter, 12:1–5.
Harris, L. (1989). S&P 500 cash stock price volatilities. The Journal of Finance, 44(5):1155–
1175.
Harvey, A. C. (1989). Forecasting, structural time series models and the Kalman filter. Cam-bridge university press.
Helske, J. (2018). KFAS: Kalman filter and smoothers for exponential family state space models.
R package version 1.3.3.
Holland, P. W. (1986). Statistics and causal inference. Journal of the American statistical Association, 81(396):945–960.
Hudgens, M. G. and Halloran, M. E. (2008). Toward causal inference with interference. Journal of the American Statistical Association, 103(482):832–842.
Imbens, G. W. and Rubin, D. B. (2015). Causal inference in statistics, social, and biomedical sciences. Cambridge University Press.
Jochum, C. and Kodres, L. (1998). Does the introduction of futures on emerging market currencies destabilize the underlying currencies? IMF Staff Papers, 45(3):486–521.
Keogh, E. and Ratanamahatana, C. A. (2005). Exact indexing of dynamic time warping.
Knowledge and information systems, 7(3):358–386.
Kim, W., Lee, J., and Kang, K. (2020). The effects of the introduction of bitcoin futures on the volatility of bitcoin returns. Finance Research Letters, 33.
Kreif, N., Grieve, R., Hangartner, D., Turner, A. J., Nikolova, S., and Sutton, M. (2016).
Examination of the synthetic control method for evaluating health policies with multiple treated units. Health economics, 25(12):1514–1528.
Kristoufek, L. (2013). Bitcoin meets Google Trends and Wikipedia: Quantifying the relation-ship between phenomena of the internet era. Scientific reports, 3(1):1–7.
Kristoufek, L. (2015). What are the main drivers of the Bitcoin price? Evidence from wavelet coherence analysis. PloS one, 10(4):1–15.
Larcker, D. F., Gordon, L. A., and Pinches, G. E. (1980). Testing for market efficiency:
a comparison of the cumulative average residual methodology and intervention analysis.
Journal of Financial and Quantitative Analysis, 15(2):267–287.
Larsen, K. (2019). MarketMatching Package Vignette. R package version 1.1.2.
Li, K. T. (2019). Statistical inference for average treatment effects estimated by synthetic control methods. Journal of the American Statistical Association.
Li, X. and Wang, C. A. (2017). The technology and economic determinants of cryptocurrency exchange rates: The case of Bitcoin. Decision Support Systems, 95:49–60.
McKenzie, M. D., Brailsford, T. J., and Faff, R. W. (2001). New insights into the impact of the introduction of futures trading on stock price volatility. Journal of Futures Markets: Futures, Options, and Other Derivative Products, 21(3):237–255.
McLannahan, B. (2014). Bitcoin exchange Mt Gox files for bankruptcy protection. The Finan-cial Times.
Meyer, B. D., Viscusi, W. K., and Durbin, D. L. (1995). Workers’ compensation and injury duration: evidence from a natural experiment. The American economic review, 85(3):322–
340.
Moln´ar, P. (2012). Properties of range-based volatility estimators. International Review of Financial Analysis, 23:20–29.
Moriarty, E. J. and Tosini, P. A. (1985). Futures trading and the price volatility of gnma certificates further evidence. The Journal of Futures Markets (pre-1986), 5(4):633–641.
Murry, J. P., Stam, A., and Lastovicka, J. L. (1993). Evaluating an anti-drinking and driving advertising campaign with a sample survey and time series intervention analysis. Journal of the American Statistical Association, 88(421):50–56.
Nakamoto, S. (2008). Bitcoin: A peer-to-peer electronic cash system.
Neslin, S. A., Henderson, C., and Quelch, J. (1985). Consumer promotions and the acceleration of product purchases. Marketing science, 4(2):147–165.
Noirjean, S., Mariani, M., Mattei, A., and Mealli, F. (2020). Exploiting network information to disentangle spillover effects in a field experiment on teens’ museum attendance. arXiv preprint arXiv:2011.11023.
86
O’Neill, S., Kreif, N., Grieve, R., Sutton, M., and Sekhon, J. S. (2016). Estimating causal effects: considering three alternatives to difference-in-differences estimation. Health Services and Outcomes Research Methodology, 16(1-2):1–21.
Papadogeorgou, G., Mealli, F., Zigler, C. M., Dominici, F., Wasfy, J. H., and Choirat, C. (2018).
Causal impact of the hospital readmissions reduction program on hospital readmissions and mortality. Preprint. Available atarXiv:1809.09590.
Pauwels, K., Hanssens, D. M., and Siddarth, S. (2002). The long-term effects of price promotions on category incidence, brand choice, and purchase quantity. Journal of marketing research, 39(4):421–439.
Rambachan, A. and Shephard, N. (2019). A nonparametric dynamic causal model for macroe-conometrics. Preprint. Available at arXivpreprintarXiv: 1903. 01637.
Robins, J. M. (1986). A new approach to causal inference in mortality studies with a sustained exposure period—application to control of the healthy worker survivor effect. Mathematical modelling, 7(9-12):1393–1512.
Robins, J. M., Greenland, S., and Hu, F.-C. (1999). Estimation of the causal effect of a time-varying exposure on the marginal mean of a repeated binary outcome. Journal of the American Statistical Association, 94(447):687–700.
Rosenbaum, P. R. (2007). Interference between units in randomized experiments. Journal of the American Statistical Association, 102(477):191–200.
Rubin, D. (1980). Discussion of” randomization analysis of experimental data in the fisher randomization test” by d. basu. Journal of the American statistical association, 75(371):591–
593.
Rubin, D. B. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of educational Psychology, 66(5):688–701.
Rubin, D. B. (1975). Bayesian inference for causality: The importance of randomization. In The Proceedings of the Social Statistics Section of the American Statistical Association, pages 233–239.
Rubin, D. B. (1978). Bayesian inference for causal effects: The role of randomization. The Annals of statistics, 6(1):34–58.
Rubin, D. B. (1981). Estimation in parallel randomized experiments. Journal of Educational Statistics, 6(4):377–401.
Rubin, D. B. (1984). Bayesianly justifiable and relevant frequency calculations for the applies statistician. The Annals of Statistics, 12(4):1151–1172.
Rubin, D. B. (2005). Causal inference using potential outcomes: Design, modeling, decisions.
Journal of the American Statistical Association, 100(469):322–331.
Ryan, A. M., Burgess Jr, J. F., and Dimick, J. B. (2015). Why we should not be indifferent to specification choices for difference-in-differences. Health services research, 50(4):1211–1235.
Salvador, S. and Chan, P. (2007). Toward accurate dynamic time warping in linear time and space. Intelligent Data Analysis, 11(5):561–580.
S¨avje, F., Aronow, P. M., and Hudgens, M. G. (2020). Average treatment effects in the presence of unknown interference. Annals of Statistics. In print.
Shi, S. (2017). The impact of futures trading on intraday spot volatility and liquidity: Evidence from Bitcoin market. Preprint. Available at https: // ssrn. com/ abstract= 3094647. Sims, C. A. (1972). Money, income, and causality. The American economic review, 62(4):540–
552.
Smith, T. C., Spiegelhalter, D. J., and Thomas, A. (1995). Bayesian approaches to random-effects meta-analysis: a comparative study. Statistics in medicine, 14(24):2685–2699.
Sobel, M. E. (2006). What do randomized studies of housing mobility demonstrate? causal inference in the face of interference. Journal of the American Statistical Association, 101(476):1398–1407.
Stein, J. C. (1987). Informational externalities and welfare-reducing speculation. Journal of political economy, 95(6):1123–1145.
Sutton, A. J. and Abrams, K. R. (2001). Bayesian methods in meta-analysis and evidence synthesis. Statistical methods in medical research, 10(4):277–303.
Sutton, A. J. and Higgins, J. P. (2008). Recent developments in meta-analysis. Statistics in medicine, 27(5):625–650.
Tchetgen, E. J. T. and VanderWeele, T. J. (2012). On causal inference in the presence of interference. Statistical methods in medical research, 21(1):55–75.
VanderWeele, T. J. (2010). Direct and indirect effects for neighborhood-based clustered and longitudinal data. Sociological methods & research, 38(4):515–544.
88
Viviano, D. and Bradic, J. (2019). Synthetic learner: model-free inference on treatments over time. Preprint. Available at arXiv:1904.01490.
West, M. and Harrison, J. (2006). Bayesian forecasting and dynamic models. Springer Science
& Business Media.
Zellner, A. and Siow, A. (1980). Posterior odds ratios for selected regression hypotheses.
Trabajos de estad´ıstica y de investigaci´on operativa, 31(1):585–603.
Zuckerman, M. J. (2018). First Bitcoin futures contract expires at $10,900, win for bears. COINTELEGRAPH, available at https://cointelegraph.com/news/
first-bitcoin-futures-contract-expires-at-10900-win-for-bears.
Appendices
A
A.1 Additional tables
Tables 13 and 14report the results of the estimated C-ARIMA models in the pre-intervention period for, respectively, store and competitor brands.
Tables 15 and 16 report the estimated marginal effect as defined in (19) and the estimated conditional effect ˆτ¯t((1, 1), (0, 1)). Tables17and18display the results of the trend and seasonal model estimated using a set of covariates with, respectively, the price difference and the price ratio.
Table19shows the results of alternative C-ARIMA models fitted on the Garman-Klass volatility proxy without regressors. Finally, Table 20reports the Pearson’s linear correlation coefficients and the Spearman’s rho of the covariates included in the Bitcoin application.
90
Table13:Storebrands.Estimatedcoefficientsand(standarderrorsinparentheses)oftheC-ARIMAmodelsfittedtothe11storebrands inthepre-interventionperiod.Thedependentvariableisthedailysalescountsofeachproductinlogscale. Dependentvariable: Item1Item2Item3Item4Item5Item6Item7Item8Item9Item10Item11 φ10.863∗∗∗0.865∗∗∗0.833∗∗∗0.496∗∗∗0.867∗∗∗0.890∗∗∗0.828∗∗∗0.949∗∗∗0.839∗∗∗0.364∗∗∗0.951∗∗∗ (0.035)(0.037)(0.047)(0.053)(0.037)(0.032)(0.042)(0.034)(0.050)(0.053)(0.039) φ20.221∗∗∗0.177∗∗ (0.052)(0.055) φ30.216∗∗∗ (0.054) θ1−0.323∗∗∗−0.340∗∗∗−0.348∗∗∗−0.251∗∗−0.458∗∗∗−0.191∗−0.509∗∗∗−0.476∗∗∗−0.841∗∗∗ (0.065)(0.074)(0.086)(0.082)(0.063)(0.080)(0.076)(0.085)(0.072) θ2−0.212∗∗ (0.074) Φ10.120∗0.113∗0.203∗∗∗0.260∗∗∗0.235∗∗∗0.234∗∗∗0.240∗∗∗0.124∗0.774∗∗∗0.688∗∗∗ (0.056)(0.057)(0.055)(0.053)(0.056)(0.057)(0.059)(0.059)(0.204)(0.146) Θ10.181∗∗−0.682∗∗−0.503∗∗ (0.058)(0.231)(0.172) c7.131∗∗∗6.767∗∗∗6.995∗∗∗7.912∗∗∗7.113∗∗∗6.535∗∗∗7.740∗∗∗7.800∗∗∗6.876∗∗∗6.312∗∗∗6.711∗∗∗ (0.167)(0.167)(0.159)(0.152)(0.072)(0.107)(0.099)(0.063)(0.025)(0.110)(0.044) price−1.721∗∗∗−1.441∗∗∗−1.656∗∗∗−2.218∗∗∗−1.433∗∗∗−1.841∗∗∗−1.423∗∗∗−2.316∗∗∗−1.727∗∗∗−2.118∗∗∗−1.175∗∗∗ (0.356)(0.355)(0.339)(0.288)(0.287)(0.304)(0.281)(0.116)(0.117)(0.176)(0.177) hol0.176∗∗∗0.165∗∗∗0.168∗∗∗0.160∗∗∗0.174∗∗∗0.162∗∗∗0.162∗∗∗0.164∗∗∗0.163∗∗∗0.076∗0.208∗∗∗ (0.033)(0.033)(0.034)(0.029)(0.027)(0.033)(0.027)(0.023)(0.026)(0.035)(0.028) Dec.Sun0.323∗∗∗0.369∗∗∗0.306∗∗∗0.292∗∗∗0.379∗∗∗0.411∗∗∗0.367∗∗∗0.327∗∗∗0.390∗∗∗0.191∗0.374∗∗∗ (0.063)(0.062)(0.069)(0.063)(0.056)(0.067)(0.054)(0.053)(0.051)(0.075)(0.072) Sat0.265∗∗∗0.243∗∗∗0.248∗∗∗0.276∗∗∗0.293∗∗∗0.277∗∗∗0.292∗∗∗0.223∗∗∗0.214∗∗∗0.135∗∗∗0.162∗∗∗ (0.024)(0.024)(0.028)(0.026)(0.022)(0.026)(0.021)(0.021)(0.020)(0.035)(0.036) Sun−1.291∗∗∗−1.355∗∗∗−1.321∗∗∗−1.213∗∗∗−1.140∗∗∗−1.261∗∗∗−1.173∗∗∗−1.203∗∗∗−1.291∗∗∗−1.398∗∗∗−1.514∗∗∗ (0.028)(0.027)(0.031)(0.029)(0.026)(0.029)(0.025)(0.025)(0.022)(0.037)(0.036) Mon−0.135∗∗∗−0.188∗∗∗−0.163∗∗∗−0.175∗∗∗−0.010−0.062∗−0.037−0.093∗∗∗−0.123∗∗∗−0.139∗∗∗−0.222∗∗∗ (0.028)(0.028)(0.032)(0.030)(0.026)(0.029)(0.026)(0.025)(0.022)(0.035)(0.036) Tue−0.225∗∗∗−0.263∗∗∗−0.271∗∗∗−0.251∗∗∗−0.170∗∗∗−0.233∗∗∗−0.207∗∗∗−0.200∗∗∗−0.226∗∗∗−0.229∗∗∗−0.305∗∗∗ (0.028)(0.028)(0.032)(0.030)(0.026)(0.029)(0.026)(0.025)(0.022)(0.035)(0.036) Wed−0.247∗∗∗−0.250∗∗∗−0.266∗∗∗−0.271∗∗∗−0.209∗∗∗−0.243∗∗∗−0.232∗∗∗−0.249∗∗∗−0.266∗∗∗−0.262∗∗∗−0.312∗∗∗ (0.027)(0.027)(0.030)(0.028)(0.025)(0.028)(0.025)(0.025)(0.021)(0.037)(0.036) Thr−0.218∗∗∗−0.218∗∗∗−0.207∗∗∗−0.239∗∗∗−0.201∗∗∗−0.211∗∗∗−0.210∗∗∗−0.234∗∗∗−0.230∗∗∗−0.249∗∗∗−0.258∗∗∗ (0.024)(0.024)(0.028)(0.026)(0.022)(0.026)(0.021)(0.021)(0.020)(0.035)(0.036) Observations386386386386386386386386386386386 σ20.0220.0220.0220.0170.0140.0220.0140.0110.0130.0270.017 AkaikeInf.Crit.-355.744-358.769-352.505-453.681-519.897-366.579-539.814-620.166-577.969-282.397-460.103 Note:·p<0.1;∗p<0.05;∗∗p<0.01;∗∗∗p<0.001
Table14:Competitorbrands.Estimatedcoefficientsand(standarderrorsinparentheses)oftheC-ARIMAmodelsfittedtothe10 competitorbrandsinthepre-interventionperiod.Thedependentvariableisthedailysalescountsofeachproductinlogscale. Dependentvariable: Item1Item2Item3Item4Item5Item6Item7Item8Item9Item10 φ1−0.442∗∗∗0.959∗∗∗0.949∗∗∗−0.090∗0.876∗∗∗0.973∗∗∗1.800∗∗∗0.939∗∗∗0.855∗∗∗0.786∗∗∗ (0.026)(0.014)(0.016)(0.043)(0.053)(0.157)(0.088)(0.028)(0.034)(0.083) φ20.349∗∗∗0.850∗∗∗−0.161∗∗−0.075−0.821∗∗∗ (0.026)(0.038)(0.054)(0.130)(0.081) φ30.915∗∗∗ (0.023) θ11.402∗∗∗0.940∗∗∗−0.408∗∗−0.768∗∗∗−0.566∗∗∗−0.239∗∗∗−0.514∗∗∗ (0.013)(0.047)(0.142)(0.109)(0.068)(0.068)(0.120) θ21.000∗∗∗−0.117. (0.014)(0.070) Φ10.108∗0.907∗∗∗0.224∗∗∗0.162∗∗0.852∗∗∗0.281∗∗∗ (0.054)(0.030)(0.054)(0.053)(0.088)(0.053) Θ1−1.000∗∗∗−0.779∗∗∗ (0.021)(0.103) c8.292∗∗∗7.862∗∗∗7.828∗∗∗8.335∗∗∗8.873∗∗∗8.380∗∗∗10.189∗∗∗7.029∗∗∗6.773∗∗∗6.327∗∗∗ (0.299)(0.297)(0.150)(0.162)(0.137)(0.127)(0.292)(0.048)(0.050)(0.108) price−2.459∗∗∗−2.350∗∗∗−2.074∗∗∗−2.193∗∗∗−2.210∗∗∗−2.257∗∗∗−3.742∗∗∗−1.490∗∗∗−1.308∗∗∗−2.499∗∗∗ (0.163)(0.124)(0.135)(0.130)(0.139)(0.121)(0.286)(0.089)(0.127)(0.220) hol0.145∗0.161∗∗∗0.099∗0.154∗∗∗0.107∗∗0.155∗∗∗0.138∗0.178∗∗∗0.131∗∗∗0.165∗∗∗ (0.059)(0.044)(0.048)(0.045)(0.033)(0.029)(0.062)(0.025)(0.029)(0.038) Dec.Sun0.411∗∗∗0.538∗∗∗0.446∗∗∗0.515∗∗∗0.431∗∗∗0.379∗∗∗0.363∗∗∗0.413∗∗∗0.338∗∗∗0.437∗∗∗ (0.093)(0.076)(0.068)(0.085)(0.057)(0.056)(0.095)(0.046)(0.063)(0.069) Sat0.323∗∗∗0.262∗∗∗0.272∗∗∗0.244∗∗∗0.322∗∗∗0.357∗∗∗0.315∗∗∗0.216∗∗∗0.263∗∗∗0.215∗∗∗ (0.036)(0.030)(0.013)(0.034)(0.024)(0.027)(0.037)(0.018)(0.025)(0.027) Sun−0.984∗∗∗−1.143∗∗∗−1.083∗∗∗−1.351∗∗∗−1.011∗∗∗−0.992∗∗∗−1.057∗∗∗−1.303∗∗∗−1.288∗∗∗−1.303∗∗∗ (0.047)(0.039)(0.018)(0.045)(0.034)(0.031)(0.050)(0.020)(0.030)(0.029) Mon−0.106∗−0.182∗∗∗−0.172∗∗∗−0.199∗∗∗−0.050−0.085∗∗0.017−0.117∗∗∗−0.119∗∗∗−0.119∗∗∗ (0.051)(0.042)(0.018)(0.048)(0.037)(0.032)(0.054)(0.020)(0.031)(0.029) Tue−0.254∗∗∗−0.240∗∗∗−0.266∗∗∗−0.267∗∗∗−0.206∗∗∗−0.241∗∗∗−0.166∗∗−0.208∗∗∗−0.231∗∗∗−0.229∗∗∗ (0.051)(0.042)(0.018)(0.048)(0.037)(0.032)(0.054)(0.020)(0.031)(0.029) Wed−0.237∗∗∗−0.222∗∗∗−0.284∗∗∗−0.285∗∗∗−0.197∗∗∗−0.254∗∗∗−0.218∗∗∗−0.237∗∗∗−0.242∗∗∗−0.270∗∗∗ (0.046)(0.039)(0.016)(0.044)(0.033)(0.030)(0.049)(0.020)(0.029)(0.028) Thr−0.208∗∗∗−0.182∗∗∗−0.220∗∗∗−0.237∗∗∗−0.187∗∗∗−0.232∗∗∗−0.200∗∗∗−0.196∗∗∗−0.219∗∗∗−0.234∗∗∗ (0.036)(0.030)(0.012)(0.034)(0.024)(0.027)(0.037)(0.018)(0.025)(0.027) Observations386386385386386386386386386386 σ20.0800.0460.0490.0440.0210.0170.0940.0130.0170.027 AkaikeInf.Crit.150.457-66.040-33.990-85.448-372.686-451.392207.605-575.002-449.421-281.026 Note:·p<0.1;∗p<0.05;∗∗p<0.01;∗∗∗p<0.001
92
Table 15: Posterior mean and 95% credible intervals of the temporal average mean marginal causal effect of the new price policy on the ten store brands computed at three time horizons. In this table, ˆ¯
τt stands for the mean marginal effect ˆτ¯t(s, 4). There is evidence of a causal effect when the credible intervals do not include zero.
Time horizon:
1 month 3 months 6 months
ˆ¯
τt 2.5% 97.5% ˆ¯τt 2.5% 97.5% ˆ¯τt 2.5% 97.5%
1 3.53 −12.27 19.33 2.39 −21.98 26.88 3.55 −32.92 39.98
2 3.55 −7.39 14.51 2.51 −15.09 19.39 3.34 −22.05 29.27
3 4.02 −7.00 16.14 2.71 −15.91 20.70 3.97 −24.13 31.25
4 24.06 2.07 49.43 11.58 −26.51 50.12 12.22 −46.22 68.82
5 1.98 −24.69 28.29 3.73 −40.92 48.08 5.74 −60.05 73.52
6 4.85 −7.97 17.53 5.94 −14.73 26.51 6.78 −24.61 38.53
7 39.19 0.04 77.11 17.33 −40.86 76.01 14.84 −74.46 102.94
8 12.67 −14.32 39.30 11.71 −34.58 54.99 8.63 −57.56 73.01
9 20.46 −9.44 50.67 8.19 −39.87 57.46 6.44 −65.70 82.99
10 6.26 0.52 11.98 4.86 −4.22 14.22 2.72 −11.39 16.84
Table 16: Posterior mean and 95% credible intervals of the temporal average conditional causal effect of the new price policy on the ten store brands computed at three time horizons. In this table, ˆ¯
τt stands for the conditional effect ˆτ¯t((1, 1), (0, 1)). There is evidence of a causal effect when the credible intervals do not include zero.
Time horizon:
1 month 3 months 6 months
ˆ¯
τt 2.5% 97.5% τˆ¯t 2.5% 97.5% ˆ¯τt 2.5% 97.5%
(1) s 0.09 −0.14 0.39 0.11 −0.14 0.38 0.12 −0.12 0.37
c −0.34 −1.68 0.75 −0.63 −1.64 0.78 −0.78 −1.55 0.64
(2) s 0.08 −0.12 0.30 0.10 −0.14 0.31 0.11 −0.10 0.29
c −0.30 −0.86 0.34 −0.54 −0.93 0.34 −0.65 −0.74 0.24
(3) s 0.11 −0.15 0.36 0.12 −0.12 0.36 0.11 −0.09 0.35
c −0.22 −0.79 0.67 −0.37 −0.94 0.58 −0.21 −0.74 0.23
(4) s 0.28 −0.97 1.50 0.50 −0.92 1.62 0.71 −0.47 4.17
c −1.04 −3.92 2.64 −2.13 −4.18 2.36 −3.22 −19.10 1.11
(5) s −0.15 −2.69 4.01 −0.12 −7.83 1.30 −0.27 −23.15 1.39
c −0.08 −2.48 2.53 −0.08 −2.73 2.67 −0.07 −2.23 3.66
(6) s 0.17 −0.28 0.57 0.12 −0.31 0.67 −0.02 −0.27 0.55
c −0.34 −1.62 0.84 −0.31 −1.90 0.73 −0.29 −1.50 0.70
(7) s 0.20 −1.16 1.60 0.21 −1.14 1.62 0.20 −1.15 1.63
c −1.09 −21.89 18.58 −1.46 −21.83 18.40 −1.26 −22.02 18.63
(8) s 0.12 −2.75 2.79 0.09 −4.46 4.22 0.18 −6.69 7.86
c −0.02 −12.99 14.38 0.15 −19.31 23.83 −0.31 −39.60 32.96
(9) s 0.64 −42.52 43.54 1.00 −70.38 78.18 0.81 −106.58 119.18
c −0.29 −45.17 44.19 −0.25 −74.86 72.15 0.28 −112.62 115.76
(10) s 0.09 −2.76 3.08 0.08 −5.16 4.39 0.13 −7.60 7.00
c 0.04 −5.83 6.93 0.07 −8.62 10.90 −0.02 −17.00 16.37
Table 17: Temporal average general causal effects of the new price policy on the ten store (s) -competitor (c) pairs computed at three time horizons. In this table, ˆ¯τt stands for the general effect ˆ¯
τt((1, 0), (0, 0)) and the results are obtained including in the set of covariates the difference in price between the store and competitor brand prior to the intervention (in the post-intervention period the difference in price is computed from the prior price).
1 month 3 months 6 months
ˆ¯
τt 2.5% 97.5% ˆ¯τt 2.5% 97.5% τˆ¯t 2.5% 97.5%
(1) s 7.86 -22.72 39.39 6.01 -44.36 54.53 8.69 -62.66 81.65
c 24.76 -101.23 154.16 18.14 -189.20 223.43 7.94 -299.89 322.01
(2) s 6.32 -15.06 27.87 4.64 -27.51 36.56 5.78 -43.30 55.55
c 14.36 -65.53 97.56 8.08 -129.50 142.40 -1.55 -206.59 198.41
(3) s 7.74 -15.37 31.07 5.76 -32.76 40.91 8.98 -45.53 64.71
c 17.60 -60.32 98.08 12.58 -116.06 142.92 6.48 -182.11 198.27 (4) s 47.39 0.94 96.95 23.29 -49.15 104.14 24.21 -88.64 136.26 c 31.44 -74.80 140.15 23.04 -156.67 205.96 14.52 -259.18 280.48 (5) s 4.51 -46.29 57.07 8.11 -75.41 91.55 13.40 -108.70 136.45 c 48.56 -55.74 160.97 18.78 -155.55 199.51 11.59 -255.06 276.53 (6) s 10.05 -14.63 35.36 12.24 -28.79 54.40 14.69 -45.35 76.51 c 25.66 -39.05 92.53 7.03 -101.58 117.02 5.53 -159.96 167.62 (7) s 80.83 6.45 158.56 38.12 -82.24 154.90 34.47 -137.44 209.06 c 184.75 -216.88 596.71 106.78 -553.29 757.07 92.10 -904.77 1086.75 (8) s 25.29 -25.76 77.12 23.02 -62.62 103.02 14.70 -111.95 135.90
c 15.27 -14.96 45.95 5.17 -44.71 53.87 3.01 -68.34 73.61
(9) s 41.09 -8.93 89.23 16.95 -61.21 99.53 13.91 -102.74 132.98 c 18.71 -30.61 71.21 2.68 -77.27 80.47 3.93 -114.88 122.98
(10) s 12.16 1.06 23.02 9.42 -8.54 26.50 5.12 -21.80 32.30
c -0.21 -8.89 8.87 1.64 -13.12 17.01 3.64 -17.52 24.97
Table 18: Temporal average general causal effects of the new price policy on the ten store (s) -competitor (c) pairs computed at three time horizons. In this table, ˆ¯τt stands for the general effect ˆ¯
τt((1, 0), (0, 0)) and the results are obtained including in the set of covariates the price ratio between the store and competitor brand prior to the intervention (in the post-intervention period the ratio is computed from the prior price).
1 month 3 months 6 months
ˆ¯
τt 2.5% 97.5% ˆ¯τt 2.5% 97.5% ˆ¯τt 2.5% 97.5%
(1) s 7.86 -23.99 40.25 5.57 -43.61 56.18 7.60 -65.59 81.24
c 24.24 -103.31 149.08 18.24 -190.07 236.58 9.94 -302.70 321.87
(2) s 6.29 -15.08 27.85 4.58 -28.01 36.62 5.78 -43.58 55.33
c 14.43 -65.19 97.88 8.04 -129.58 142.52 -1.94 -206.72 198.81
(3) s 7.69 -15.61 31.11 5.69 -33.02 41.00 8.94 -45.58 65.00
c 17.67 -60.31 98.22 12.55 -116.11 142.85 6.40 -182.21 198.30 (4) s 47.59 -1.43 95.37 23.49 -52.91 99.97 26.11 -85.00 143.55 c 30.86 -76.21 142.37 21.79 -156.22 203.90 12.56 -247.89 285.21 (5) s 4.93 -45.95 56.46 8.44 -74.91 93.47 13.63 -107.63 138.26 c 48.63 -58.86 160.54 18.78 -161.04 203.72 11.66 -267.79 280.47
(6) s 9.89 -14.74 34.85 12.05 -29.01 54.04 14.37 -46.42 75.06
c 25.76 -38.76 92.99 7.05 -100.67 117.62 5.59 -155.74 167.47 (7) s 80.67 1.53 161.11 36.73 -84.22 156.80 31.45 -150.41 207.70 c 183.01 -222.65 583.47 108.84 -559.14 799.66 102.14 -892.35 1113.15 (8) s 23.54 -28.05 73.80 22.06 -59.32 103.49 14.64 -113.07 140.54
c 14.98 -15.50 44.80 4.46 -44.03 53.53 2.35 -69.75 75.51
(9) s 41.00 -7.02 87.54 16.93 -64.31 97.09 14.35 -106.63 136.62 c 18.68 -32.60 69.15 2.66 -82.03 83.46 4.81 -113.13 120.65
(10) s 12.50 1.45 23.71 9.62 -9.64 27.65 5.07 -23.35 31.58
c -0.11 -9.77 9.72 1.72 -13.10 16.31 3.77 -18.52 25.31
Table 19: Estimated coefficients (standard errors in parentheses) of alternative C-ARIMA models fitted on Garman-Klass volatility proxy (in log scale) up to the day before each intervention. In this table, ˆτ(m) indicates the estimated contemporaneous effect of each announcement; ˆτ is the average¯ contemporaneous effect; ˆ¯τ(CBOE) is the temporal average pointwise effect of the CBOE future; and, ˆ¯
τt(CM E) is the temporal average pointwise effect of the CME futures at 1-week, 2-weeks and 3-weeks horizons (indicated with t = 7, t = 14 and t = 21).
Dependent variable:
Ann.1 Ann.2 Ann.3 Ann.4 CBOE CME
φ1 1.223∗∗∗ 1.214∗∗∗ 1.224∗∗∗ 1.222∗∗∗ 1.223∗∗∗ 1.214∗∗∗
(0.059) (0.057) (0.056) (0.056) (0.059) (0.057)
φ2 −0.256∗∗∗ −0.246∗∗∗ −0.254∗∗∗ −0.252∗∗∗ −0.256∗∗∗ −0.246∗∗∗
(0.051) (0.049) (0.048) (0.049) (0.051) (0.049)
θ1 −0.776∗∗∗ −0.774∗∗∗ −0.783∗∗∗ −0.781∗∗∗ −0.776∗∗∗ −0.774∗∗∗
(0.047) (0.045) (0.044) (0.044) (0.047) (0.045)
c −3.804∗∗∗ −3.769∗∗∗ −3.745∗∗∗ −3.742∗∗∗ −3.804∗∗∗ −3.769∗∗∗
(0.100) (0.099) (0.102) (0.102) (0.100) (0.099)
ˆ
τ(m) −0.52 −0.05 0.43 −0.32
(0.51) (0.50) (0.50) (0.50)
ˆ¯
τ −0.11
(0.25) ˆ¯
τ(CBOE) −0.24
(0.38) ˆ¯
τt=7(CM E) 0.80∗∗
(0.38) ˆ¯
τt=14(CM E) 0.78∗∗
(0.37) ˆ¯
τt=21CM E 0.71∗
(0.37)
Observations 1,153 1,243 1,274 1,277 1,153 1,243
σ2 0.257 0.252 0.254 0.255 0.257 0.252
Akaike Inf. Crit. 1,710.446 1,820.362 1,877.940 1,883.982 1,710.446 1,820.362
Note: ·p<0.1;∗p<0.05;∗∗p<0.01;∗∗∗p<0.001
Table20:Pearson’slinearcorrelationcoefficient(uppertriangular)andSpearman’srho(lowertriangular)betweenGarman-Klassvolatility(GK)andthe covariatesincludedintheanalysis,computedintheperiodbeforethefirstintervention. GKeurusdvolmxwovolmxefvolshcvoleurusdmxwomxefshcm1inflationgdpmidratehashtot.btchalv GK0.06-0.04-0.030.020.02-0.010.010.01-0.000.000.010.020.01-0.01-0.00 eurusdvol0.040.470.430.350.03-0.03-0.05-0.02-0.030.01-0.000.05-0.04-0.07-0.16 mxwovol-0.060.440.670.270.02-0.10-0.11-0.07-0.03-0.04-0.000.04-0.04-0.06-0.28 mxefvol-0.050.410.630.370.00-0.07-0.09-0.08-0.00-0.02-0.050.06-0.02-0.01-0.22 shcvol0.020.370.300.39-0.03-0.05-0.09-0.120.02-0.01-0.04-0.03-0.010.04-0.55 eurusd0.010.020.020.00-0.01-0.03-0.04-0.100.000.00-0.05-0.01-0.020.010.05 mxwo0.04-0.01-0.10-0.04-0.020.010.650.19-0.040.010.010.05-0.01-0.020.04 mxef0.02-0.02-0.08-0.06-0.09-0.040.520.36-0.02-0.010.030.040.03-0.030.06 shc0.000.01-0.03-0.01-0.04-0.060.120.26-0.020.040.05-0.00-0.020.01-0.01 m1-0.01-0.04-0.030.010.030.05-0.05-0.010.01-0.03-0.00-0.00-0.020.120.01 inflation0.000.00-0.01-0.04-0.030.020.04-0.020.02-0.04-0.030.00-0.020.02-0.01 gdp0.000.000.00-0.05-0.05-0.08-0.030.030.04-0.01-0.05-0.00-0.010.01-0.01 midrate0.000.060.060.07-0.04-0.000.030.04-0.02-0.000.00-0.01-0.08-0.000.02 hash0.02-0.03-0.02-0.01-0.02-0.01-0.000.00-0.04-0.030.01-0.01-0.080.08-0.00 tot.btc-0.00-0.06-0.07-0.020.020.020.02-0.010.040.11-0.00-0.01-0.020.32-0.00 halv-0.01-0.20-0.32-0.25-0.580.040.030.08-0.04-0.020.02-0.010.05-0.010.03